GHS and other inequalities

We use a transformation due to Percus to give a simple derivation of the Griffiths, Hurst, and Sherman, and some other new inequalities, for Ising ferromagnets with pair interactions. The proof makes use of the Griffiths, Kelly, and Sherman and the Fortuin, Kasteleyn, and Ginibre inequalities.Work supported in part by USAFOSR-73-2430.     

A simple proof of the GHS and further inequalities

We formulate and prove a general set of correlation inequalities for spin — 1/2 Ising ferromagnets with pair interactions. One of these is the Griffiths-Hurst-Sherman inequality. The proof is obtained using Gaussian random variables.     

New inequalities for Ising ferromagnets

It is shown that for Ising ferromagnets which obey the Lee-Yang theorem the Ursell functions or cumulants of the magnetization variable at nonzero external field satisfy series of inequalities. Several relations connecting Ursell functions with nonzero and zero field are derived.     

Correlation inequalities and the decay of correlations in ferromagnets

We prove a variety of new correlation inequalities which bound intermediate distance correlations from below by long distance correlations. Typical is the following which holds for spin 1/2 nearest neighbor Ising ferromagnets: $$\langle S_\alpha S_\gamma \rangle \leqq \sum\limits_{\delta \in B} {\langle S_\alpha S_\delta \rangle } \langle S_\delta S_\gamma \rangle$$ whereB is any subset of the lattice whose removal divides the lattice into pieces with α,γ in distinct components. We describe various applications, e.g. the above inequality implies the critical exponent inequality η<1.     

Surface tension,percolation, and roughening

We describe inequalities relating to the interface between coexisting phases of Ising ferromagnets. Some implications for the nature of the roughening transition are discussed.Supported by National Science Foundation Grant No. MCS78-01885 at Princeton University and No. PHYS78-15920 at Rutgers University.Part of this work was done while Jean Bricmont was at the Mathematics Department of Princeton University and Joel L. Lebowitz at the Institute for Advanced Study, Princeton.     

A phenomenological theory of metastable states in disordered Ising magnets

A mechanism of the formation of an exponentially large number of metastable states in magnetic phases of disordered Ising magnets as a result of condensation of fractal delocalized modes near the localization threshold is suggested. The thermodynamic properties of metastable states are studied in the effective-field approximation in the vicinity of transitions in magnets with zero uniform magnetization in the ground state such as dilute antiferromagnets, spin glasses, and dilute ferromagnets with dipole interaction. These properties are shown to determine the parameters of nonequilibrium processes in the glassy phase, namely, the shape of the hysteresis loop, the thermodynamic values in field-cooled and zero-field-cooled regimes, and the thermoremanent and isothermal remanent magnetization values.     

Hyperscaling inequalities for percolation

A set of critical exponent inequalities for independent percolation which saturate under the hyperscaling hypothesis is proved. One of the consequences of the inequalities is the lower boundd _C6 for the upper critical dimension. The proof is based on a rigorous version of the finite size scaling argument which extends easily to other systems such as Ising ferromagnets.Supported by NSF Grant PHY-85-15288-A01     

On limit theorems for the bivariate (magnetization,energy) variable at the critical point

The limiting (magnetization, energy) bivariate variable is studied for Ising ferromagnets at the critical point. The factorization property of the limiting bivariate moment generating function is shown to be intimately connected to critical point exponent inequalities and to the behaviour of the scaling limit near and at the critical point. The validity of this can be deduced from the study of the second and the fourth magnetization cumulants at zero external field. The limiting bivariate variable is exactly calculated at the critical point for the Curie-Weiss model (MF) and for the edge of a two-dimensional Ising ferromagnet wrapped on a cylinder. It is shown that the mean field case leads to a non-Gaussian limiting distribution in contradistinction with the particular Ising model we consider for which we obtain a product of two Gaussian probability distributions.     

Return to return point memory

We describe a new class of systems exhibiting return point memory (RPM), different from those discussed before in the context of ferromagnets. We show numerically that one-dimensional random Ising antiferromagnets have exact RPM when evolving from a large field, but not when started at finite field, unlike the ferromagnetic case. This implies that the standard approach to understanding ferromagnetic RPM will fail for this case. We also demonstrate RPM with a set of variables that keeps track of spin flips at each site. Conventional RPM for the spins is a projection of this result, suggesting that spin flip variables might be a more fundamental representation of the dynamics. We also present a mapping that embeds the antiferromagnetic chain in a two-dimensional ferromagnet, and prove RPM for spin-exchange dynamics in the interior of the chain with this mapping.     

A general central limit theorem for FKG systems

A central limit theorem is given which is applicable to (not necessarily monotonic) functions of random variables satisfying the FKG inequalities. One consequence is convergence of the block spin scaling limit for the magnetization and energy densities (jointly) to the infinite temperature fixed point of independent Gaussian blocks for a large class of Ising ferromagnets whenever the susceptibility is finite. Another consequence is a central limit theorem for the density of thesurface of the infinite cluster in percolation models.     

二维磁性系统和准一维磁性纳米管的内能

采用量子统计理论的多体格林函数法计算二维单离子各向异性海森伯铁磁体、反铁磁体以及单壁铁磁纳米管的内能,对比铁磁体和反铁磁体的结果.在相同的参量下,反铁磁能量总是低于相应的铁磁能量(相变点除外).由于反铁磁能量随温度上升的速度较铁磁能量快,当温度升高到居里点T_C和奈尔点T_N时(T_C=T_N),铁磁能量和反铁磁能量相等.横向关联效应对系统内能的影响较大,不能忽略.     

Phenomenology of current-induced dynamics in antiferromagnets

We derive a phenomenological theory of current-induced staggered magnetization dynamics in antiferromagnets. The theory captures the reactive and dissipative current-induced torques and the conventional effects of magnetic fields and damping. A Walker ansatz describes the dc current-induced domain-wall motion when there is no dissipation. If magnetic damping and dissipative torques are included, the Walker ansatz remains robust when the domain wall moves slowly. As in ferromagnets, the domain-wall velocity is proportional to the ratio between the dissipative torque and the magnetization damping. In addition, a current-driven antiferromagnetic domain wall acquires a net magnetic moment.     

Role of the spin anisotropy of the interchain interaction in weakly coupled antiferromagnetic Heisenberg chains

In quasi-one-dimensional(q1D) quantum antiferromagnets, the complicated interplay of intrachain and interchain exchange couplings may give rise to rich phenomena. Motivated by recent progress on field-induced phase transitions in the q1D antiferromagnetic(AFM) compound YbAlO_3, we study the phase diagram of spin-1/2 Heisenberg chains with Ising anisotropic interchain couplings under a longitudinal magnetic field via large-scale quantum Monte Carlo simulations,and investigate the role of the spin anisotropy of the interchain coupling on the ground state of the system. We find that the Ising anisotropy of the interchain coupling can significantly enhance the longitudinal spin correlations and drive the system to an incommensurate AFM phase at intermediate magnetic fields, which is understood as a longitudinal spin density wave(LSDW). With increasing field, the ground state changes to a canted AFM order with transverse spin correlations. We further provide a global phase diagram showing how the competition between the LSDW and the canted AFM states is tuned by the Ising anisotropy of the interchain coupling.     

Upper bounds for Ising model correlation functions

A Griffiths correlation inequality for Ising ferromagnets is refined and is used to obtain improved upper bounds for critical temperatures. It is shown that, for non-negative external fields, the mean field magnetization is an upper bound for the magnetization of Ising ferromagnets.On leave (1970–71) from Northwestern University, Evanston, Illinois 60201. Supported at IAS by a grant from the Alfred P. Sloan Foundation.     

Antiferromagnets in a field and ferromagnets in a random field

A new mean field approach for antiferromagnets in a field is shown to be equivalent to a mean field treatment of ferromagnets in a random field. Phase diagrams of the two systems are analogous.     

Approximation to the susceptibilities of three-dimensional antiferromagnets

Approximate susceptibilities of simple cubic S=∞ Heisenberg and $\text{S=}\text{1}\text{2}$ Ising antiferromagnets are calculated by using the new expansion method reported previously for the case of a square-plane S=∞ Heisenberg antiferromagnet. The present method gives a good approximation to the susceptibilities of both of the Heisenberg and Ising antiferromagnets above the Néel temperature.     

Correlation inequalities for a class of even ferromagnets

We present rigorous correlation inequalities for connectedn-point functions in a class of even ferromagnets. The class includes spin-1/2 Ising models and scalar field models with potential functionV which is even and continuously differentiable withV convex on [0, ). These inequalities are obtained by pushing ahead with the method of Ellis, Monroe, and Newman at its maximum.     

Critical dynamics of magnets

We review our current understanding of the critical dynamics of magnets above and below the transition temperature with focus on the effects due to the dipole-dipole interaction present in all real magnets. Significant progress in our understanding of real ferromagnets in the vicinity of the critical point has been made in the last decade through improved experimental techniques and theoretical advances in taking into account realistic spin-spin interactions. We start our review with a discussion of the theoretical results for the critical dynamics based on recent renormalization group, mode coupling and spin-wave theories. A detailed comparison is made of the theory with experimental results obtained by different measuring techniques, such as neutron scattering, hyperfine interaction, muon spin resonance, electron spin resonance, and magnetic relaxation, in various materials. Furthermore we discuss the effects of dipolar interaction on the critical dynamics of three-dimensional isotropic antiferromagnets and uniaxial ferromagnets. Special attention is also paid to a discussion of the consequences of dipolar anisotropies on the existence of magnetic order and the spin-wave spectrum in two-dimensional ferromagnets and antiferromagnets. We close our review with a formulation of critical dynamics in terms of nonlinear Langevin equations.     

Some exact distributions of order parameter in antiferromagnetic and ferromagnetic media

Some exact dynamic and static solutions of Landau–Lifshitz equations in antiferromagnets and ferromagnets found in this work. These solutions can be used for describing distribution of order parameter in small magnetic particles.     

The low-temperature behavior of disordered magnets

We map out the low-temperature phase diagrams of dilute Ising ferromagnets and predominantly ferromagnetic ferrites, obtaining nonperturbative and essentially optimal conditions on the density of ferromagnetic couplings required to maintain long-range order. We also study mappings of dilute antiferromagnets in a uniform field onto random field ferromagnets.For the randomly dilute systems, we prove that ferromagnetically ordered states exist at low temperature if the density of ferromagnetic couplings exceeds the (appropriately defined) percolation threshold, thereby extending the result of Georgii to three or more dimensions. We also show that, for these systems, as the temperature tends to zero, the magnetization approaches the percolation probability of the corresponding Bernoulli system. In two dimensions, we prove that low-temperature ordering persists in the presence of antiferromagnetic impurities if the ferromagnetic couplings percolate and if the density of antiferromagnetic couplings is bounded above by the order of the inverse square of the corresponding percolation correlation length. For these systems, we rigorously compute the first order decrease in the zero-temperature nominal spontaneous magnetization, in terms of derivatives of the percolation probability, thereby establishing the existence of ferrimagnetically ordered states. Finally, we introduce a model of a random ferrite which exhibits spontaneous magnetization anticorrelated with the boundary conditions.National Science Foundation Postdoctoral Research Fellows. Work supported in part by the National Science Foundation under Grant No. PHY-8203669Work supported in part by the National Science Foundation under Grant No. MCS-8108814 (A03)